High-fidelity qutrit entangling gates for superconducting circuits

Ternary quantum information processing in superconducting devices poses a promising alternative to its more popular binary counterpart through larger, more connected computational spaces and proposed advantages in quantum simulation and error correction. Although generally operated as qubits, transmons have readily addressable higher levels, making them natural candidates for operation as quantum three-level systems (qutrits). Recent works in transmon devices have realized high fidelity single qutrit operation. Nonetheless, effectively engineering a high-fidelity two-qutrit entanglement remains a central challenge for realizing qutrit processing in a transmon device. In this work, we apply the differential AC Stark shift to implement a flexible, microwave-activated, and dynamic cross-Kerr entanglement between two fixed-frequency transmon qutrits, expanding on work performed for the ZZ interaction with transmon qubits. We then use this interaction to engineer efficient, high-fidelity qutrit CZ† and CZ gates, with estimated process fidelities of 97.3(1)% and 95.2(3)% respectively, a significant step forward for operating qutrits on a multi-transmon device.


Reviewer #2 (Remarks to the Author):
In the manuscript by Goss et. al., the authors demonstrated and benchmarked an all-microwave qutrit CZ gate using differential AC Stark shifts. This paper has many impressive aspects including a detailed experimental characterization of the cross-Kerr dynamics, and a clever use of echoes to construct the qutrit CZ gate without fine tuning the cross-Kerr dynamics. This work is an extension of differential AC Stark shits in qubit systems [ref. 39-42]. The ability to generate 3-dimensional Bell states using a single qutrit CZ gate is attractive; potentially allowing one to generate 3-dimsional multipartite entangled GHZ states in transmons. This work deserves publication, but I'd like to see the following questions addressed first.
1. Can the authors comment on how the charge noise in the 2 state impact the qutrit CZ gate? In the device data given in the supplementary, the T2 times for 12 and 02 levels are considerably lower than that of 01 levels.
2. I did not find a centralized description of the gate parameters for the qutrit CZ/CZ^\dagger gate. The gate times are in the caption of Fig 3, and the powers in the supplementary. What was the drive frequency and how were the frequency and power chosen?
3. Did the authors simulate terms beyond the 4 in the cross-Kerr Hamiltonian? In regions where the differential AC Stark fails the cross-Kerr Hamiltonian is presumably no longer a good approximation and other terms will be present. Fig. 2e, are there additional regions where the qutrit CZ gate does not perform well? For instance near the middle between omega_01,a and omega_01,b I would expect CZ dynamics to get complicated due to 2-photon transitions. 5. In the paragraph below equation 4, the 2 approximated equations seem to imply \alpha_11 + \alpha_22 ~ \alpha_12 + \alpha_21 ~ 0. Can the authors double-check this?

Beyond the 4 resonances present in
6. In the same paragraph the authors state "the ramp feature can lead to offsets … approximate relationship on the \alpha_ij terms with some adjustment necessary", what is the adjustment? 7. It might be helpful to have different labels for the phases \phi_a \phi_b and powers in Fig.2a  The authors implement High fidelity qutrit entangling gate that can provide the basis for qutritbased quantum computers. This gate is implemented using a cross-Kerr resonant pulse. The gate designed appears to have higher entangling power than a Cinc or Cex gate.he work does establish significance for the field. It provideds a proof of principle demonstration of a powerful entangling gate for qutrits. This demonstration of the gate likely can be used to help develop more efficient compilers for qubit architechture simulations such as those provided in Gokhale et al. "Asymptotic improvements for quantum circuits via qutrits." This work appears to be better than 82% fidelity result for a Cinc gate implemented by Morvan et al. in 2020.
I find the authors work novel and timely and warrants publication following the minor revisions.
Requested Revisions: I have a few revisions that I believe are necessary before approving for publication 1.) I would like to see a slightly wider discussion of the benefits qutrit architecture may apply to quantum algorithms. As the current connection to the benefits of qutrit hardware to algorithms seem sparse.
2.) I would be interested for the authors to comment or speculate on why they see the substantive improvement in entangling gate versus the fidelity found in [28]. Could any of this be related to the choice of using cycle benchmarking versus interleaved randomized benchmarking? 3.) Is there a reference to Weyl twirling? I would like to see it included.

High-Fidelity Qutrit Entangling Gates for Superconducting Circuits: Reviewer Response
Reviewer #1 (Remarks to the Author): This manuscript reports on high-fidelity qutrit entangling gates for superconducting transmon circuits. A microwave-activated cross-Kerr interaction is used to realize the qutrit entangling gates. The reported results constitute a good demonstration of logical operations on superconducting qutrits.
We thank the reviewer for spending the time to carefully read and provide comments on our manuscript.
My biggest reservation about the manuscript is that the main achievement is quantitative rather than qualitative. Some of the authors of this manuscript have previously reported qutrit entangling gates in references 28 and 32. In [28], the lowest infidelity was 11.1%. In this manuscript the lowest infidelity is 2.7%. This is an improvement. However, it is a quantitative improvement. It is true that the gate mechanism is different. However, this is not a major advance in the field.
On this point, we respectfully disagree with the reviewer. We would argue one of the main achievements presented in this work is the introduction and characterization of the differential-AC Stark shift for a tunable qutrit entanglement. Although the driven two-qubit-ZZ term generated by the differential AC Stark shift has been previously studied in a number of sources [refs. 39-42], the higher cross-Kerr terms generated by such an interaction have not. Employing this interaction to engineer an entangling gate with fidelity x4 better than the previous best attempt on a superconducting platform does constitute a quantitative improvement, but was only made possible due to the introduction of this new method of qutrit entanglement generation. Therefore, the gate mechanism in our work is not just different due to the new pulse scheme, but also due to the underlying physics. Prior attempts at this gate employed the parasitic, always-on dispersive coupling between two transmons to generate qutrit entangling phases; this former approach was both inflexible (as it cannot be tuned and is generally of the order~100 KHz) and relied on an interaction (quantum crosstalk) that one would like to suppress in future qutrit systems. Our work allows for an entangling interaction in qutrits with a significant on/off ratio; we have added comments about this point to the introduction to help make this clearer. In our work, we are able to generate a maximally entangling gate on a 9 dimensional Hilbert space with a competitive fidelity to the state of the art 3 qubit gates, which operate on an 8 dimensional Hilbert space. The gates presented in this work employ a pulse scheme and tune up procedure that is not more complicated than many comparable qubit gates such as the Toffoli, which has many implementations that require using the qutrit levels anyways (such as e.g. refs. 34,37, and https://arxiv.org/abs/2109.00558). For these reasons, it is the feelings of the author that this work indeed constitutes a major advance for the practical implementation of qutrit processing with superconducting circuits..
Another concern is that qutrit logic is a niche research area. Most research in the field is focused on qubits because of obvious advantages such as simplicity and better coherence.
We agree with the reviewer that most of the research in the field is dedicated to studying qubits, as they are traditionally simpler to operate and easier to fabricate with high coherence. Nonetheless, we would like to point out our {|1>,|2>} subspace coherence is already better or comparable to some qubits devices (i.e. the {|0>,|1>} subspace) in the literature and available from industry (see e.g. Fig. 1 in ref. 31 or https://qcs.rigetti.com/qpus). Moreover, high-fidelity single qutrit operations have now been demonstrated in a number of works which we cite in the manuscript. So far, however, it is not clear that utilizing the third level and operating the system as a qutrit is advantageous compared to the simplicity of working with qubits. It is our feeling, though, that our work allows tuning up and operating qutrit entangling gates in a much simpler manner, opening the door to future efforts benefiting from the advantages of achieving control over a larger Hilbert space without added hardware or on chip overhead. It is our hope that this will help catalyze the already growing subfield of quantum information processing user qutrits.
Overall the manuscript is clearly written. However, there are some poorly written parts. Here are some examples. It is not explained what "more connected" means. The work is motivated by saying that realizing multi-qudit systems is challenging. However, if transmons are qudits, then any multi-transmon circuit is a multi-qudit system. So this motivation sounds weak. The manuscript says, without a proper justification, that the interaction in this work "generates entanglement between the entire two-qutrit Hilbert space". The poor wording aside, there is no explanation for how this is different from previous work on two-qutrit gates. These problems can be fixed with a rewriting of the poorly written parts.
We thank the reviewer for these useful comments on the wording within the manuscript. We have updated the language within the manuscript to reflect our comments here. In the following, we respond point by point: -By "more connected" we mean that within a fixed quantum processor topology (i.e. a ring, a line, etc.) the amount of connected states for that processor is much higher in a qutrit processor rather than a qubit processor. For example, if we consider a line of 4 transmons with nearest neighbor coupling, then the graph of connected states with nearest neighbor coupling looks as follows: Similarly, using that same processor as qutrits, we will realize the following graph: If we want to compare the same number of nodes in the same type of topology in both cases, then we can consider 3 qubits connected linearly to 2 qutrits (6 nodes in both cases) in which case we find 9 edges for the qutrits and 8 for the qubits. We also note that this point of a more connected structure for qutrits was also made in ref. 28 [28,48]." What is meant here is that in the case of the cross-resonance effect for qutrits, the conditional Rabi oscillations are largely constrained to take place within a two level subspace of the qutrit, as can be noted from, e.g. Fig. 2 a and b in ref. [28]. In the case of the differential AC Stark shift, one can realize dynamic entangling phases simultaneously placed on the |11>, |12>, |21>, and |22> states of the two qutrits. This is sufficient as a native entangling interaction to efficiently generate maximally-entangled qutrit C-phase gates as we performed in this work.
A technical point about the qutrit CZ gate is that it does not reduce to a qubit CZ gate if we look only at the states 0/1. So the basis for generalizing the CZ gate from qubits to qutrits warrants some justification. When equation 3 is introduced, it is not clear if it is taken from the literature or introduced for the first time in this manuscript.
We thank the reviewer for allowing us to make this discussion clearer in the text. The qutrit CZ comes directly from the generalization of the Pauli group employed in qubit based computation, to the Heisenberg-Weyl group employed for qudit/qutrit computation. Using this basis allows one to generalize the Clifford group as well as the Pauli group to qudits. This generalization of the qubit Pauli group to qutrits is discussed in Supplementary Note 3

of this work. The CZ gate follows directly from the definition of the Z gate in ternary based computation. For an additional reference, see equation 8 in ref 28. Notably, the CZ and CZ^dag gate performed here are locally equivalent to the CSUM (controlled sum) gate, which is the qutrit extension of the CNOT gate in qubits. We have updated the supplementary note 3 to additionally explain the unitary of the CZ and CZ^dag gate and reference the supplementary note at this point of the main text.
Except for these issues, the manuscript reports good results in the area of superconducting qutrits.
We gratefully thank Reviewer #1 for their time and helpful comments on our manuscript. We hope that our response here and updates to the manuscript help alleviate any lingering concerns on the part of the reviewer about the manuscript.

Reviewer #2 (Remarks to the Author):
In the manuscript by Goss et. al., the authors demonstrated and benchmarked an all-microwave qutrit CZ gate using differential AC Stark shifts. This paper has many impressive aspects including a detailed experimental characterization of the cross-Kerr dynamics, and a clever use of echoes to construct the qutrit CZ gate without fine tuning the cross-Kerr dynamics. This work is an extension of differential AC Stark shits in qubit systems [ref. 39-42]. The ability to generate 3-dimensional Bell states using a single qutrit CZ gate is attractive; potentially allowing one to generate 3-dimsional multipartite entangled GHZ states in transmons. This work deserves publication, but I'd like to see the following questions addressed first.
We gratefully thank Reviewer #2 for their time, helpful comments, and appreciation of our work contained within the manuscript. Here we respond point by point to the questions the reviewer would like to see addressed, and have updated the manuscript to reflect any relevant changes and clarifications discussed here.
1. Can the authors comment on how the charge noise in the 2 state impact the qutrit CZ gate? In the device data given in the supplementary, the T2 times for 12 and 02 levels are considerably lower than that of 01 levels.
In the case of, for example, the chip in 3. Did the authors simulate terms beyond the 4 in the cross-Kerr Hamiltonian? In regions where the differential AC Stark fails the cross-Kerr Hamiltonian is presumably no longer a good approximation and other terms will be present.

Our master equation simulation included 4 levels for both of the transmons. In general, we tried to perform the interaction in regions where we expect the cross-Kerr
Hamiltonian to be a good approximation of the dynamics in our system by maintaining a large detuning from the resonances plotted in Figure 2. However, we do expect that transient two level systems and higher level transitions such as two photon transitions do impact our data. We discuss this further in Supplementary Note 8, but in general, we only included data that fit well to the linear accumulation of entangling phases one would expect under a cross-Kerr evolution.

4.
Beyond the 4 resonances present in Fig. 2e, are there additional regions where the qutrit CZ gate does not perform well? For instance near the middle between omega_01,a and omega_01,b I would expect CZ dynamics to get complicated due to 2-photon transitions.
This is correct, two photon transitions would be a concern in this region. Thankfully, due to the flexible nature of this entangling interaction, we have the freedom to always place our Stark drive away from points of concern such as this. In other words, thanks to the flexibility of this entangling scheme, we do not have to perform the gate in such a region of concern.
We are sorry for the confusion, this was poorly worded in the original manuscript. Here, the two conditions on the cross-Kerr parameters correspond to two different conditions for two different gates, not two conditions that are simultaneously met.  Figure 3, and adjusting the pulse parameters until the target entangling phases are most accurately met. We have expanded this discussion in the main text to add clarity to this point.
7. It might be helpful to have different labels for the phases \phi_a \phi_b and powers in Fig.2a and 3, to be consistent with Eq. 2 and Fig. 1.
Thank you for pointing out this inconsistency, we have corrected this in the text. I find the authors work novel and timely and warrants publication following the minor revisions.
We gratefully thank Reviewer #3 for their time, helpful comments, and appreciation of our work contained within the manuscript. Here we respond point by point to the questions the reviewer would like to see addressed, and have updated the manuscript to reflect any relevant changes and clarifications discussed here.
Requested Revisions: I have a few revisions that I believe are necessary before approving for publication 1.) I would like to see a slightly wider discussion of the benefits qutrit architecture may apply to quantum algorithms. As the current connection to the benefits of qutrit hardware to algorithms seem sparse.
We thank the reviewer for the suggestion to expand this discussion in the text. We have updated the main text to make more explicit the benefits of qutrit hardware to algorithms.
2.) I would be interested for the authors to comment or speculate on why they see the substantive improvement in entangling gate versus the fidelity found in [28]. Could any of this be related to the choice of using cycle benchmarking versus interleaved randomized benchmarking?
The substantive improvement in the gate fidelity when compared to ref.
[28] is due to the breakthrough of the differential AC-Stark shift that allows one to tune up the effective diagonal coupling (ZZ for qubits or cross-Kerr for qutrits as introduced in this work) between two transmons. This allowed us to significantly speed up the time of the two-qutrit gate by a factor of 3 and to reduce the number of single qutrit pulses required to implement the proper unitary. This driving scheme was not known when ref. [28] was published. We also attribute this significant increase to the higher quality of coherence in our two devices. This was possible thanks to improvements in the fabrication of our device detailed in https://iopscience.iop.org/article/10.1088/1361-6668/ab8617. Finally, we want to point out to the reviewer that in ref. [32], the fidelity of the 2 qutrit gate was measured with Cycle Benchmarking as well. In general, Cycle Benchmarking provides a tighter bound on the fidelity of entangling gates than interleaved randomized benchmarking; additionally, to our knowledge, two qutrit randomized benchmarking has not been performed due to the large overhead in the required number of multi-qutrit gates per circuit.
3.) Is there a reference to Weyl twirling? I would like to see it included.
Thank you for pointing out that this reference was omitted in the main text. Weyl twirling was first experimentally performed in ref. 32 "Qutrit Randomized Benchmarking", where it was referred to as Pauli twirling (as the Weyl group is just the qutrit generalization of the qubit Pauli group). We have updated the text to include this reference.